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  <h1 class="w3-xxxlarge">Group Explorer 3.0 Help: First Tutorial</h1>
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                <h3>Contents</h3>
                <ul>
                
                    <li><a style="text-decoration: none;"
                          href="#first-tutorial">First Tutorial</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#pictures">Pictures</a></li>
                
                    <li><a style="text-decoration: none;"
                          href="#features">Features</a></li>
                
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                <h3><a style="text-decoration: none;"
                      href="http://nathancarter.github.io/group-explorer">Group Explorer Home</a></h3>

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                <h3>Other help pages</h3>
                
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                        <li><a href=".."
                               style="text-decoration: none;"
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                               >Main Help Page</a></li>
                    
                    
                    
                        <li><a href="../welcome/"
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                               >Welcome</a></li>
                    
                    
                    
                        <li><a href="../gs-index/"
                               style="text-decoration: none;"
                               class=""
                               >Getting Started</a></li>
                    
                    
                    
                    
                    
                        <li><a href="../tu-index/"
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                               >Tutorials</a></li>
                    
                    
                    
                    
                    
                        <li><a href="../rf-um-index/"
                               style="text-decoration: none;"
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                               >User Manual</a></li>
                    
                    
                    
                    
                    
                    
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            <h2 id="first-tutorial">First Tutorial</h2>
<p>Welcome to the first <em>Group Explorer</em> tutorial! This first tutorial is
designed to give you both motive and opportunity to begin your own
explorations. We start with a tour of the major visuals <em>Group Explorer</em>
offers and then have a description of the most useful features.</p>
<p>This page is meant to be a springboard, so don&rsquo;t hesitate to follow any of
the (many) links below, even if you haven&rsquo;t read this whole page. That&rsquo;s the
point of this tutorial&ndash;to help you start to explore.</p>
<h2 id="pictures">Pictures</h2>
<p><em>Group Explorer</em> gives four ways to answer the question &ldquo;What does a group
look like?&rdquo;  This is its primary aim and the reason for its creation.</p>
<p>The multiplication table is the simplest way to picture a group, and
explicitly shows <em>the group operation.</em></p>
<p><a href="http://nathancarter.github.io/group-explorer/Multtable.html?groupURL=groups/S_3.group"><img alt="Multiplication table for S_3" src="../s_3_multtable_unlabelled.png" /></a>
&nbsp; &nbsp; &nbsp;
<a href="http://nathancarter.github.io/group-explorer/Multtable.html?groupURL=groups/A_4.group"><img alt="Multiplication table for A_4" src="../a_4_multtable_thumbnail.png" /></a></p>
<ul>
<li><a href="../rf-groupterms/#multiplication-table">Definition of a multiplication table</a></li>
<li><a href="../gs-mt-intro/">Introduction to multiplication tables</a> (Getting Started pages)</li>
<li><a href="../tu-mt-manip/">Basic manipulations of a multiplication table</a> (Tutorial)</li>
<li><a href="../rf-um-mt-options/">Complete information about the multiplication table interface</a> (User Manual)</li>
</ul>
<p>CITE(VGT-4.3 MM-1.5 DE-5.3)</p>
<p>Groups describe symmetry, sometimes the symmetry of three-dimensional
objects.  Thus these objects have <em>the symmetry the group describes.</em></p>
<p><a href="http://nathancarter.github.io/group-explorer/SymmetryObject.html?groupURL=groups/Z_9.group"><img alt="Cycle described by Z_9" src="../z_9_symmetry.png" /></a>
&nbsp; &nbsp; &nbsp;
<a href="http://nathancarter.github.io/group-explorer/SymmetryObject.html?groupURL=groups/A_5.group"><img alt="Platonic solid described by A_5" src="../a_5_symmetry.png" /></a></p>
<ul>
<li><a href="../rf-groupterms/#objects-of-symmetry">Definition of an object of symmetry</a></li>
<li><a href="../rf-um-os-options/">Complete information about the object of symmetry interface</a> (User Manual)</li>
</ul>
<p>CITE(VGT-3 MM-1.3 TJ-12.2)</p>
<p>Cayley diagrams show the group as a graph of elements interacting with each
other.  They show the group&rsquo;s <em>structure</em> very well, exemplifying Cayley&rsquo;s
theorem by showing <em>how the group acts on itself.</em></p>
<p><a href="http://nathancarter.github.io/group-explorer/CayleyDiagram.html?groupURL=groups/Z_3.group"><img alt="Cayley diagram of Z_3" src="../z_3_cayley_unlabelled.png" /></a>
&nbsp; &nbsp; &nbsp;
<a href="http://nathancarter.github.io/group-explorer/CayleyDiagram.html?groupURL=groups/A_4.group"><img alt="Cayley diagram of A_4" src="../a_4_cayley_unlabelled.png" /></a></p>
<ul>
<li><a href="../rf-groupterms/#cayley-diagrams">Definition of a Cayley diagram</a></li>
<li><a href="../gs-cd-intro/">Introduction to Cayley diagrams</a> (Getting Started pages)</li>
<li><a href="../tu-cd-manip/">Basic manipulations of a Cayley diagram</a> (Tutorial)</li>
<li><a href="../rf-um-cd-options/">Complete information about the Cayley diagram interfacel</a> (User Manual)</li>
</ul>
<p>CITE(VGT-2.4 MM-1.2 DE-3)</p>
<p>Cycle graphs show an important relationship among the elements of the group
by exhibiting <em>the group&rsquo;s orbits.</em></p>
<p><a href="http://nathancarter.github.io/group-explorer/CycleGraph.html?groupURL=groups/V_4.group"><img alt="Cycle graph of V_4" src="../v_4_cycle_unlabelled.png" /></a>
&nbsp; &nbsp; &nbsp;
<a href="http://nathancarter.github.io/group-explorer/CycleGraph.html?groupURL=groups/Z_16.group"><img alt="Cycle graph of Z_16" src="../z_16_cycle_unlabelled.png" /></a></p>
<ul>
<li><a href="../rf-groupterms/#cycle-graph">Definition of a cycle graph</a></li>
<li><a href="../rf-um-cg-options/">Complete information about the cycle graph interface</a> (User Manual)</li>
</ul>
<p>CITE(VGT-5.1)</p>
<h2 id="features">Features</h2>
<p>Here are some of hte most important things not to miss about how <em>Group
Explorer</em> makes gropu theory more accessible and intuitive.</p>
<h3 id="each-picture-of-a-group-is-very-interactive">Each picture of a group is very interactive</h3>
<p><img alt="A cycle graph with highlights" src="../illustration-cychigh1.png" /></p>
<p><a href="../tu-discovery/">Study</a> properties of the group by reorganizing a diagram
or <a href="../rf-um-subsetlistbox/#highlight-subset-by-submenu">highlighting it in different ways</a>.  Save useful
views as images to use in a document, email, or website.</p>
<h3 id="group-explorer-has-a-large-library-of-groups"><em>Group Explorer</em> has a large library of groups</h3>
<p><img alt="A screenshot of the group library" src="../illustration-grouptable.png" /></p>
<p><a href="http://nathancarter.github.io/group-explorer/GroupExplorer.html">The main page of the
application</a>
is a list of groups known to the software.  It is very handy for learning by
example, for testing conjectures, and getting one&rsquo;s hands dirty in the real
subject matter of group theory.  Each group in the library, when clicked,
opens a <a href="../rf-um-groupwindow/">group info window</a> full of useful information
about the group.</p>
<h3 id="help-links-appear-in-many-locations">Help links appear in many locations</h3>
<p><img alt="A screenshot of some help links" src="../illustration-helplinks.png" /></p>
<p>Many portions of the software contain links to useful explanations.  In
addition to the help pages you&rsquo;re reading now, <a href="../rf-um-groupwindow/">group info
windows</a> may contain links to the help documentation
or to dynamic generations of illustrations of various concepts.  Also, each
<a href="../rf-geterms/#visualizers">visualizer</a> has a <a href="../rf-um-largewindow/#help">help
button</a> to take you directly to the help for that
visualizer.</p>
<h3 id="groups-can-be-connected-and-compared">Groups can be connected and compared</h3>
<p><img alt="A screenshot of a morphism" src="../illustration-morph5.png" /></p>
<p>You need not be content with examining groups in isolation.
<a href="../tu-sheets/">Sheets</a> are an important tool that lets you create
<a href="../rf-um-morphedit/">homomorphisms</a> between groups, see complex diagrams
like subgroup lattices, and more.</p>
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